{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bayesian Data Analysis, 3rd ed\n",
    "##  Chapter 6, demo 1\n",
    "\n",
    "Posterior predictive checking demo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import os, sys\n",
    "# add utilities directory to path\n",
    "util_path = os.path.abspath(os.path.join(os.path.pardir, 'utilities_and_data'))\n",
    "if util_path not in sys.path and os.path.exists(util_path):\n",
    "    sys.path.insert(0, util_path)\n",
    "\n",
    "# import from utilities\n",
    "import plot_tools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# edit default plot settings\n",
    "plt.rc('font', size=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# data\n",
    "data_path = os.path.abspath(\n",
    "    os.path.join(\n",
    "        os.path.pardir,\n",
    "        'utilities_and_data',\n",
    "        'light.txt'\n",
    "    )\n",
    ")\n",
    "y = np.loadtxt(data_path)\n",
    "# sufficient statistics\n",
    "n = len(y)\n",
    "s2 = np.var(y, ddof=1)  # Here ddof=1 is used to get the sample estimate.\n",
    "s = np.sqrt(s2)\n",
    "my = np.mean(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Create 9 random replicate data sets from the posterior predictive density.\n",
    "# Each set has same number of virtual observations as the original data set.\n",
    "replicates = np.random.standard_t(n-1, size=(9,n)) * np.sqrt(1+1/n)*s + my"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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vkr4y/WT6zSSPTu/IdXGSk6vqU+k/pXpG+reCYa21VlUvTu9Y9a30vgIL4y6v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\nAADDBAoAYJhAAQAMEygAgGECBQAw7P8DoA2iNMHZ6AUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3f604510f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot them along with the real data set in random order subplot\n",
    "fig, axes = plt.subplots(5, 2, sharex=True, sharey=True, figsize=(9, 12))\n",
    "fig.subplots_adjust(top=0.95, wspace=0.4)\n",
    "order = np.random.permutation(10)\n",
    "for i, ax in enumerate(axes.flat):\n",
    "    ax.hist(\n",
    "        replicates[order[i]] if order[i] < 9 else y,\n",
    "        np.arange(-45, 55, 5)\n",
    "    )\n",
    "    plot_tools.modify_axes.only_x(ax)\n",
    "axes[0, 0].set_xlim([-50, 58])\n",
    "fig.suptitle(\n",
    "    \"Light speed example: Observed data + Replicated datasets.\\n\"\n",
    "    \"Can you spot which one is the observed data?\"\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Compare the minimum of the real data set into the minimum of a replicated dataset.\n",
    "\n",
    "The distribution of the minimum value of a replicated data set can be \n",
    "calculated analytically. Consider $n$ samples of $X_i$, where $X_i$ has cumulative distribution function $F(x)$ and probability distribution function $f(x)$. The cumulative distribution function of the minimum of the $n$ samples is $1 - (1 - F(x))^n$ and the probability distribution function is its derivative $n f(x) (1 - F(x))^{n-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Calculate the pdf of the minumum of a replicated dataset\n",
    "x = np.linspace(-60, 20, 150)\n",
    "pdf = stats.t.pdf(x, df=n-1, loc=my, scale=np.sqrt(s2*(1+1/n)))\n",
    "cdf = stats.t.cdf(x, df=n-1, loc=my, scale=np.sqrt(s2*(1+1/n)))\n",
    "pdf_min = n * pdf * (1 - cdf)**(n-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5iq7FixeXG41GQWBgYMrll18+8oYbbnA6LVtXUxF2NX1ed9MGEtJXNG0cIYR4\nCJo2jhBCvBwVc0II8QJUzAkhxAtQMSeEEC/QXTG3WK1WGhhLCCGMtdVipzfR666Y/1JYWKgymUyi\nrka9EEIIGThWq5WrrKz0B+D0/kZdXploNpsfrKurW9jQ0DCP53kNqFuGEEJYsAI4YTab5ztboMtx\n5oQQQjwDtbQJIcQLUDEnhBAvQMWcEEK8ABVzQgjxAlTMCSHEC1AxJ4QQL0DFnBBCvAAVc0II8QJU\nzAkhxAtQMSeEEC9AxZwQQrwAFXNCCPECVMwJIcQLUDEnhBAvQMWcEEK8QJeTU/QS3Ri9h6ZMmQIA\n2Lt3L9MchBC30C9Tc1LLnBBCvAAVc0II8QL92c1Cemjp0qWsIxBCvEx/zgFKfeaEENJ71GfuqTIz\nM5GZmck6BiHEi1DLnAEazUII6YBa5oQQQmyomBNCiBegYk4I6bF+7JYl/YyKOSGkWydK6vHAhsMY\n/bcfsPtUOes4xAE6AcrAgQMHAAATJ05knISQ7m05fAHPfpENf6kIQQoxzlU2YsmMkZh/5VDW0bxF\nv5wApWJOCHGqztCCKa/txfAQBT74wzj4CDg8sSUT358sxxcLJ2JstJp1RG9Ao1k81YEDB9pb54S4\nszW7c6FvbsXfbk2EUiKCzNcHr9+VAqXEBx/tP886HumALudn4PnnnwdA48yJe8uraMTG3wpx9/go\njAxTtj8uF/vgnvFR+OCX8yipa4ZWJWWYkthRy5wQ4tCHv+RD7CPAU9cPv+i5uRNjAACfHChwbSji\nFBVzQshFTGYLvsnSYVpiKAL8xBc9r1VJcWNSKDYfuoAmk5lBQtIZFXNCyEX25lRCbzTjtpRwp8vc\ne3kUGoxm7M+rcmEy4gwVc0LIRb7KLEWA3BeTYwOdLjMuWgOpSIhfqJi7BToBysCaNWtYRyDEqQZj\nK3afLsfdl0XCR+i8vefrI8DlQzX4JZeKuTugljkDKSkpSElJYR2DEIe+P1kOk9mK21K13S47OTYQ\n+VVNKKlrdkEy0hUq5gzs3r0bu3fvZh2DEId+Ol2OMH8JUiNV3S57ZVwQAGA/tc6Zo2LOwPLly7F8\n+XLWMQi5iMXK48C5alwZFwiO6/7CxOEhfghSiPEz9ZszR8WcENIuu6Qe9c2tmNTFic+OOI7D5NhA\n7M+rgtVKd/RgiYo5IaSdfZhhT4s5YOs3r2lqwZmyhoGKRXqAijkhpN3PuZVICFMi0MGFQs6MabvZ\nVnZJ3UDWY7IhAAAfR0lEQVTFIj1AxZwQAgAwtJhxrLAOk+N63ioHgGiNDH5iH5wo0Q9QMtITNM6c\ngffee491BEIucuh8DVos1i4vFHJEIOCQEK7EidL6AUpGeoKKOQPx8fGsIxBykQPnquErFOCyGE2v\nX5sU7o/Nhwphtli7vNCIDBza6gzs3LkTO3fuZB2DkN85XFCD0RH+kPoKe/3aURFKGFutyK9qGoBk\npCeomDOwevVqrF69mnUMQtqZzBacLNH3eeagpHB/ALa5QgkbVMwJIThRokeLxYrUqL4V86FBfpCI\nBHQSlCEq5oQQHCusBQCMie7+En5HhAIOCWF0EpQlKuaEEBy7UItIjRTBCkmf3yNJ649TpXq6EpQR\nKuaEDHI8z+NoYS3G9rGLxS4p3B+NJjMKawz9lIz0Bg1NZGDjxo2sIxDSrqSuGRUNpvYrOfvKPunz\nGZ0eQwLl/RGN9AIVcwYiIyNZRyCk3VF7f/kltsyHBtkK+LnKxkvORHqPulkY2LJlC7Zs2cI6BiEA\ngIwLdZCKhBgRqrik95GLfRDuL8G5ShprzgK1zBlYv349ACA9PZ1xEkKA48V1GKX175crN4cF+1HL\nnBFqmRMyiJktVpzW6ZGk9e+X9xsW5IdzFY3geRrR4mpUzAkZxPIqG2FstWJ0RD8V82A/NLVYUKY3\n9sv7kZ6jYk7IIJZdbLvIp/9a5m0nQSuo39zVqJgTMohll9RD7ivE0H4aShgb5AeARrSwQCdAGdi2\nbRvrCIQAsBXzxHB/CATdT97cE0EKMRQSH+RVUDF3NWqZMxAYGIjAwN5NAEBIfzNbrDhVqseofuov\nB2wTPA8LohEtLFAxZ2DDhg3YsGED6xhkkMutaITJbMWofuovt6NizgYVcwaomBN3kF3Svyc/7YYF\ny1GuN6HB2Nqv70u6RsWckEHqRD+f/LSznwTNpytBXYqKOSGD1MlSPRLClf128tPOfo+W/CrqanEl\nKuaEDEJWK4/TOj0Sw/u3iwUAItQyAMCF6uZ+f2/iHBVzQgahwhoDDC0WJLTdtrY/SURChColKKyh\nbhZXonHmDHz77besI5BB7lSpba7OhPD+L+YAEBUgQxFNUuFS1DJnQCaTQSaTsY5BBrHTOj18BBxi\ng/0G5P2jNTIUVlMxdyUq5gy88847eOedd1jHIIPYKZ0escF+kIiEA/L+URoZKhpMaG6xDMj7k4tR\nMWdg69at2Lp1K+sYZBA7Vapvn+ZtIEQF2L55FtVS69xVqJgTMshUN5pQpjcOyMlPuyiNfUQLFXNX\noWJOyCBzWtcAYOBOfgJAdIBtrHkhnQR1GSrmhAwyp3S2y/gHsptFLRNBIfbBhWoanugqVMwJGWRO\nleoR5i+BRu47YOvgOA6RGhkuUMvcZWicOQN79+5lHYEMYmfKGjAiVDHg64kOkCGnvGHA10NsqGVO\nyCDSYrYir6JxQLtY7KI0MhTXNMNqpcmdXYGKOQOvvfYaXnvtNdYxyCB0rrIRZiuPeBe0zKMCZGix\nWGlyZxehYs7A119/ja+//pp1DDIInSmzXcbvqpY5ALoS1EWomBMyiJwpa4CvUIAh/XwPc0eiNbZ1\n0D1aXIOKOSGDyBldA2KD/SASDvyffqi/BBwHFNfRrXBdgYo5IYPImTK9S0ayAICvjwAhCglKaqmY\nuwINTWRAKpWyjkAGodqmFpTrTRgR5ppiDgBatRQlddTN4gpUzBnYtWsX6whkEDpTZhvzPSJ04E9+\n2mlVUmQU1bpsfYMZdbMQMkjYR7K4smUeoZZCV2eEhcaaDzgq5gwsW7YMy5YtYx2DDDJndA3QyH0R\n5Cd22Tq1ainMVh7lNNZ8wFExZ+Cnn37CTz/9xDoGGWTsJz85jnPZOrUq2/mhEhrRMuComBMyCFis\nPM6WN7q0vxywdbMAoBEtLkDFnJBB4EKNAc2tFpf2lwOAVmW7CrSYZhwacFTMCRkEzujaTn66aIy5\nndRXiAC5L3WzuAANTWQgICCAdQQyyJwua4CAA+KCXVvMAdtJ0GLqZhlwVMwZ+OKLL1hHIINMTpke\nMYFySH2FLl+3ViWl+5q7AHWzEDIInClrwEgXn/y0i1BLUVLbDJ6nseYDiYo5A8899xyee+451jHI\nINFkMqOw2uDy/nI7rUoKk9mKqsYWJusfLKibhYFff/2VdQQyiNi7OFwxIYUjWrVtREtJXTOCFK67\nYGmwoZY5IV4up+2eLK6YkMIR+4VDNDxxYFExJ8TLndHp4Sf2aS+qrqZtu3ColIYnDigq5oR4udNl\nDRge4geBwHWX8XeklPjAT+yD0jq6P8tAoj5zBiIiIlhHIIMEz/M4o9PjluRwZhk4jkOYv4Ra5gOM\nijkDmzZtYh2BDBKl9UbojWaMYNRfbheukkJXTy3zgUTdLIR4Mftl/CMZjWSxC1dRy3ygUTFnYNGi\nRVi0aBHrGGQQsM8uxGpYol24vxTVTS0wtlqY5vBm1M3CQGZmJusIZJA4pdMjUiOFQiJimiO8bSSN\nrt6IIYFyplm8FbXMCfFiZ3R6l9/D3JEwlQQADU8cSFTMCfFSxlYLzlc1MbtYqCP7GHcq5gOHijkh\nXupseQOsPPuTnwAQ6m9vmdOIloFCfeYMDB8+nHUEMgic0bG9jL8jsY8QgX5iapkPICrmDLz//vus\nI5BB4HSZHlKREFEaGesoAACtSoLSeirmA4W6WQjxUqd1esSHKphdxt9ZmL+UWuYDiIo5Aw899BAe\neugh1jGIF+N53jYhhRt0sdjZrwKlSSoGBnWzMHD27FnWEYiXK9MbUWdoxcgw9ic/7cJVEhhaLKhv\nboVK5ss6jtehljkhXsh+8tMdxpjb2S8cKqGulgFBxZwQL3S6zHZPlhFu1DIPaxueqKPhiQOCijkh\nXui0rgFalRRKxpfxd9R+4RCNaBkQ1GfOQEpKCusIxMud0end6uQnAAT6iSEScnTh0AChYs7AmjVr\nWEcgXszYakF+VRNuTAplHeV3BAIOoTRJxYChbhZCvExeRSMsVt6tTn7ahdNY8wFDxZyBOXPmYM6c\nOaxjEC912j4hhRud/LSjGYcGDnWzMFBcXMw6AvFiZ8oaIBEJEB3gfvcND1dJUKY3wmLlIXSTK1O9\nBbXMCfEyp3V6xIco3LJYhquksFh5VDRQ67y/UTEnxIvwPI/TbjiSxS7cn+5rPlComBPiRUrrjag1\ntCJR6886ikP/uwqUWub9jfrMGbjiiitYRyBe6kRJPQAgKdw9W+b26eN01DLvd1TMGVixYgXrCMRL\nnSiph1DAuW03i1IigkLsQ90sA4C6WQjxIidK6hEb5AeJSMg6ilPhKilKaXhiv6NizkBaWhrS0tJY\nxyBe6ESpHola92yV24Wp6CrQgUDdLAxUV1ezjkC8UIXeiMoGE0a56clPu3CVFFnF9axjeB1qmRPi\nJU6Utp38dPdi7i9BTVMLmlssrKN4FSrmhHiJEyV6cBzc9uSnXTjdCndAUDEnxEucKKnHkEA5/MTu\n3XtqL+Y0SUX/cu+97qWuvfZa1hGIFzpRUo9xMRrWMbpFV4EODCrmDLzwwgusIxAvU91oQmm9EUlu\nPpIFAEL8xeA46mbpb9TNQogXOFlqu+1tUrh7n/wEALGPEIF+YmqZ9zMq5gxMnz4d06dPZx2DeBH7\nSJZEDyjmQNuFQ9Rn3q+omDPQ3NyM5mZqlZD+c7JEj0iNFP4y95nAuStalYS6WfoZFXNCvMCJ0nqP\n6GKxC2ubPo7nedZRvAYVc0I8XH1zKwqrDW5/sVBH4SopjK1W1BlaWUfxGlTMCfFwJz3kys+Owv1t\nt8ItoZOg/YaGJjJw8803s45AvMjJEttIlkQ3vYe5I+1XgdY1e9SHkDujYs7An//8Z9YRiBc5UVqP\nMH8JAv3ErKP0WPtVoHQr3H5D3SyEeLgTJfUeMyTRLkDuC1+hgMaa9yMq5gxMmTIFU6ZMYR2DeIEG\nYyvyq5o84srPjgQCznZfc2qZ9xsq5oR4sOzievA8kBKpYh2l18L8aZKK/kTFnBAPllFUB8Azi7nt\nKlAq5v2FijkhHiyzqA5DAuVQyXxZR+k1rUqKcr0RZouVdRSvQMWcEA/F8zwyi+qQHOFZJz/twvyl\nsPJAeYOJdRSvQEMTGbjrrrtYRyBeQFdvm/PTE7tYACBcZbtwSFfXDG3bUEXSd1TMGXjkkUdYRyBe\nINPeXx6lZpykb+xjzUvqmjGOcRZvQN0sDBgMBhgMBtYxiIfLLKqDr1CAkWEK1lH6JKztkn66FW7/\noJY5AzNmzAAA7N27l20Q4tEyi+qQEK6E2EfIOkqfKCQi+EtFNKKln1DLnBAPZLZYkV1c77H95XYR\naimKa+lban+gYk6IBzqta0BzqwVjoj2zv9xOq5KiuJZa5v2BijkhHuhoYQ0AYJyHF/MItQzFtTRJ\nRX+gYk6IBzpSWIswf0n7iBBPFaGWornVgpqmFtZRPB6dAGVg3rx5rCMQD3e0sBZjPbxVDtiKOQAU\n1zYjwINu4euOqJgzQMWcXIqSumbo6o0e38UC2LpZAFsxT/bwk7msUTcLA1VVVaiqqmIdg3ioIwVt\n/eUxGsZJLp1Wbb9wiEa0XCpqmTMwa9YsADTOnPTNscJayHyFGBHqmRcLdeQvFUEp8aERLf2AWuaE\neJgjhbVIjVLBR+gdf772ES3k0njH0UDIINFoMuO0To+x0Z7fxWJHFw71DyrmhHiQwwU1sPLA5UO8\nqZjTWPP+QMWcEA9yML8GIiGHMR56p0RHItRSGFosqDW0so7i0egEKAMLFy5kHYF4qN/yq5ESqYLU\n1zNvruXI/8aaG6CRe96MSe6CijkD6enprCMQD9RoMiO7pB6PTBnGOkq/so81L6ltxugIGmveV9TN\nwkBRURGKiopYxyAe5khBDSxWHhOGBrCO0q+0Ha4CJX1HLXMG7rvvPgA0zpz0zm9e2F8O/G+seRGN\naLkk1DInxEMcPF+N5Ajv6i+3iw6Qo7CaivmloGJOiAdoNJmRVVzvdV0sdlEBMhRWN7GO4dGomBPi\nAQ7mV8Ni5TFxmHcW82iNbay52WJlHcVjUTEnxAP8nFsFiUiAsTHe1V9uFx0gg9nKQ1dPkzv3FZ0A\nZeCpp55iHYF4mJ9zKzFhaIDHTt7cnegAOQCgoLoJkRoZ4zSeiYo5A7fccgvrCMSDlNQ141xlE+4Z\nH8U6yoCJDrAV8MJqA66MYxzGQ1E3CwM5OTnIyclhHYN4iF9yKwEAVw0PYpxk4IQoJPD1EeBCDY1o\n6StqmTOwYMECADTOnPTMz7lVCFGKERfsxzrKgBEIOERpZCioohEtfUUtc0LcmMXK45e8KkyODQLH\ncazjDKiYABm1zC8BFXNC3Fh2ST3qDK24angg6ygDLkpju3CIboXbN1TMCXFjP50uh4ADrvbi/nK7\n6AAZmlstqGwwsY7ikaiYE+LGfjpdgXHRGqhk3n9r2Cj7iBbqaukTOgHKwNKlS1lHIB5AV9+MUzo9\nnps+gnUUl4hpG2teWG3AZTHeM5OSq1AxZ+C6665jHYF4gJ9OVwAArh0ZzDiJa2hVUgg44ALdo6VP\nqJuFgczMTGRmZrKOQdzcf85UIDpAhmFB3jsksSNfHwEi1DLk0/DEPqGWOQOLFi0CQOPMiXPNLRbs\nz6vC7MujvH5IYkfDguQ4V0nFvC+oZU6IG9p3tgImsxXXjwxhHcWlhgX5Ib+yERYrDU/sLSrmhLih\nb7PLoJH7YvyQwXUiMDbYDyazFaV1NIVcb1ExJ8TNGFst+Ol0OaYlhsBHOLj+RIe13bIgr6KRcRLP\nM7iOFEI8wL6zlWhqsWDGqDDWUVwutu1k77lKKua9RSdAGXjllVdYRyBubFe2DmqZyGuniOuKWu4L\njdyXWuZ9QMWcgYkTJ7KOQNyUsdWC3acrcNOoMIgGWReLXWyQH7XM+2BwHi2MHThwAAcOHGAdg7ih\nvTmVaDSZMWP04OtisRsW7Ect8z6gljkDzz//PAAaZ04u9uWxYgQpxJjkpRM398SwIDlqDa2oaWqB\nRu7996TpL9QyJ8RN1DS1YE9OBWamhA+6USwdxdKIlj4ZvEcMIW7m66xStFp43DEmgnUUpobRiJY+\noWJOiJv44lgJRoYpMTJMyToKU1qVFBKRgFrmvUTFnBA3kFfRgONFdUgbo2UdhTmBgENcsAI5ZQ2s\no3gUOgHKwJo1a1hHIG5m88EiiIQcbkuhYg4ACWFK7D5dDp7nB9WNxi4FtcwZSElJQUpKCusYxE00\nt1iw7WgRpiWGIkghZh3HLSSEK1Hd1IJyPU0h11NUzBnYvXs3du/ezToGcRM7s0qhN5px34Ro1lHc\nRkK47bzBKV094ySeg7pZGFi+fDkAmnGI2Hz6WyGGh/gNujskdsV+EvhkiR5TRwyu2wD3FbXMCWEo\nq7gOx4vrce/l0dQ33IGf2AcxATKc0ulZR/EYVMwJYej9/+ZDIfbB7TSK5SKJ4f44WUrFvKeomBPC\nSFGNAd9m6zB7QhSUEhHrOG4nIVyJCzUG6I2trKN4BCrmhDDywc/5EAo4/HHSENZR3FJCW7/5GR2N\nN+8JOgHKwHvvvcc6AmGspqkFW44UYWaKFiFKCes4bimxbUTLydJ6OjncA1TMGYiPj2cdgTD2wc/5\nMJmteOiqoayjuK0ghRiBfr7Ub95D1M3CwM6dO7Fz507WMQgjlQ0m/HN/AW4ZHY64EAXrOG6L4ziM\njlAh40It6ygegVrmDKxevRoAcMsttzBOQlh4Z28eWixWPHH9cNZR3N64GDX+c6YC1Y0mBPjR1bFd\noZY5IS5UWteMT3+7gFljIjAkUM46jtu7LMbWV360kFrn3aFiTogLrfruDMABj14byzqKRxil9Yev\nUIAjVMy7RcWcEBc5XFCD7ZmlWHDVUESoZazjeASJSIjkSH8cLqhhHcXtUTEnxAUsVh4v7jiJMH8J\nFk4ZxjqORxkXo8GJkno0t1hYR3FrVMwZ2LhxIzZu3Mg6BnGhTw8W4pROj+dnjITMl8Yd9MZlMWq0\nWngcL65jHcWtUTFnIDIyEpGRkaxjEBcpqjFg5a4zuDIuEDePDmMdx+OMjbKdBD18nrpaukLFnIEt\nW7Zgy5YtrGMQF7BaeTyzLQsCjsPKtNF0Z8Q+8JeJEB+iwEEq5l2iYs7A+vXrsX79etYxiAts/K0Q\nv+ZXY8lNI6FVSVnH8VhXxwfh4PlqNNBNt5yiYk7IAMkqrsPL35zG1BHBuPsy6la7FNeNDEGrhcd/\nz1axjuK2qJgTMgDqDa145NNjCFKIsfrOZOpeuURjolRQy0TYfbqcdRS3RcWckH7WarHi/z47hrJ6\nI9bNToVa7ss6ksfzEQpwzYhg/OdMBcwWK+s4bomKOSH9iOd5vLD9BH7OrcLLtydhTJSadSSvcf3I\nENQ3t9LVoE7QgFcGtm3bxjoCGSBv/pSLzw8X4f+uiUX6ZVGs43iVK4cHwVcowO5T5ZgwNIB1HLdD\nLXMGAgMDERgYyDoG6Wdv78nDmt25SBsTgaduoDsi9jc/sQ8mxQbgm2wddbU4QMWcgQ0bNmDDhg2s\nY5B+wvM83tydi1e/z8HtqVqsmkXjyQdK+mVR0NUb8Z8zFayjuB2O5/n+eq9+eyNvN2XKFADA3r17\nmeYgl85ssWLp9hP4/HAR0sZEYNWs0RAKqJAPFLPFisl/34PhoQp88sfxrOP0l345YKhlTkgfVTea\nMO+fh9v7yF+7kwr5QPMRCnD3+Ej892wlCqqaWMdxK1TMCemDQ+drcNPaX3CooAarZo3Gn6fFU9eK\ni9wzPgpCAYfNhy6wjuJWqJgT0gvNLRa8tPMU0t//FWKRAP9+ZCLuGkdXd7pSiFKCG5NC8elvhaho\nMLKO4zaomBPSAzzP4+usUlz3+j58tP887psQjW8fuxKJ4f6sow1KT98QjxaLFau+y2EdxW3QCVAG\nDAYDAEAmo9lm3B3P8/g5twprdp/FsQt1GBmmxF9vScDlNM6ZuRW7TuO9ffnY/qdJSIlUsY5zKfql\nf46KOSEOmMwWfJutw4b9BTheXI9wfwkevTYOd42LpJOcbqLRZMY1r+1FiFKMbQ9PhEQkZB2pr6iY\ne6p33nkHAPDII48wTkI6slp5ZBbX4avMUnx1vBQ1TS0YGiTHA5OHYNbYCIh9PLZYeK0fT5XjoY1H\ncGNiKN6ePQYCz/ygpWLuqWicufuob27FL7lV+M+ZCuw7W4Gqxhb4+ghwfUII0sdFYnJsoKcWiEHj\ng5/zsfyb07h/UgxeuCnBE/dXvwSme7OQQaO5xYJzlY04XlyHzAt1yCyqQ15lI3ge8JeKcPXwIEwd\nEYxrRgTDXypiHZf00AOTh6C4thn/3F+AvIpGvJGegkA/MetYLkctcwaoZT4wjK0W1DS1QFdvhK6+\nGbo6I0rqmpFf1YRzFY0oqWtuX1YtEyElUoWUSDUmxQYgJVIFHyEN7vJUPM9j86EL+NvOU5D7CjH3\nihjMvSIaAZ5R1KmbxVPZi/mePXt+97ijXdH5IUf7y9GGd/xeDl7bg73W1/fiAVh5HmYLD7PVCovV\n/m8eFqsVrRbe9piVh9libXucR6vF9pyhxYzmFgsMLRY0t1pgaDHb/t1iQVOLBXWGFtQ0taDO0Ipa\nQwsMLZaLMsl9hRgSJMewIL/2nyStElEaGV3k44XOlOnx2vdnsft0OQQckBypwuVDAjAsSI4ojQxK\nqQgKiQ8UYhFkYiGEHAeOA+tjwTuK+Y+nyvH45xm/f6MeFI+eFD5nD/a0EF28TN8LaUdlmxcDAEJn\nr+x6QfI7Ag6Q+fpA6iuE3FcIlcwXGrkvVDIRNDJfqOW+UMt8EeYvQZhKgjB/KZQSH9Z/qISBvIoG\n7Mgsxf68KmQV18Ns7fqPkuMAAcdB0FbYOdj+39ND5/5JMXh62oi+xnWvYs5x3HcA+npf10AA7ji5\nn7vmAtw3G+XqHcrVO96Yq4rn+RsvNUB/tsz7HoLjjvA8P451js7cNRfgvtkoV+9Qrt6hXM7RGR9C\nCPECVMwJIcQLuEsxf591ACfcNRfgvtkoV+9Qrt6hXE64RZ85IYSQS+MuLXNCCCGXgIo5IYR4AebF\nnOO4uzmOO81xXBPHcec4jruyw3PXchx3huM4A8dxeziOi3ZRpr0cxxk5jmts+8np9PxsjuMK2zJv\n5zhO44pcHdYf15Zvkzvk4jhuE8dxOo7j9BzHneU4bn6n512+HzmOE3Mc92Hb9mjgOC6T47jprHO1\nrff/OI47wnGcieO4DQ6eZ5Krbd0ajuP+3XYMFXIcN9tV6+6Uw+k2YrjfujymWO43ALarGln9ALge\nQCGACbB9sGgBaNueCwRQD+BOABIArwL4zUW59gKY7+S5RAANAK4C4AdgM4DPXbzdfgDwM4BN7pCr\nbd3itn+PAFAGYCzL/QhADuCvAGLajq2b27ZPjBscX3cAmAlgPYANnZ5jlqtt/Z8B2NJ2DE1uy5Lo\nqvV3t40Y7zenxxTr/cbzPPNifgDAA06eewjAgU4bshnACBfk6qqYvwJgc4f/DwPQAkDhom12N4Ct\nbQfVJnfJ1WG98QB0AO5ivR8dZMsCkOYuuQAsd1DMWR738rZjZniHxzYCWOnqfeVsG7nDfnN0TLlD\nLmbdLBzHCQGMAxDEcVwex3HFHMe9xXGctG2RRADH7cvzPN8E4Fzb466wguO4Ko7j9nMcN6XD451z\nnUPbH8BAB+I4TgngJQBPOniaWa62bO9wHGcAcAa2Yv6tk1yu3o/2fCGwbYuT7pTLAZa5hgMw8zx/\ntsNjx1207p5ym/3W6Zhinotln3kIABGAWQCuBJACIBXA0rbn/WD72tJRPQCFC7I9C2AobN0+7wPY\nyXHcMDfItQzAhzzPFzt4jmUu8Dz/SNu6rgTwJQCTO+QCAI7jRAA+BfAxz/Nn3CWXEyxz+QHQM1p3\nT7nFfnNwTDHPNWDFvO0kIu/k5xfYvoIAwDqe53U8z1cBeB3AjLbHGwEoO72tErY+qoHMBZ7nD/I8\n38DzvInn+Y8B7Gedi+O4FADXAXjDyVsw2152PM9beJ7/BUAEgIXukIvjOAFsXQUtAP6vw1sw315O\nDEiuHmK57p5intHJMcU814DNNMTz/JTuluE4rhi/v4tsx3+fBPCHDsvKYesHPolL0JNcjl6G/92m\n8iSA5A65hgIQAzjr4HX9lovjuEWwnWi5wNnuy+kHQMhxXALP82NY5XLCB7Z9BTDcj5xtQ30I27fA\nGTzPt3Z42p2Or44GJFcPnQXgw3FcHM/zuW2PJbto3T3Fcvt0dUwxzQWA+QnQlwAcBhAMQA3bCI1l\nbc8FwfY1JQ22s8N/h2tGQagATGtbpw+AewE0oe2kEGx9YHrYuhPkADbBBaNGAMgAhHb4eQ3ANgBB\njHMFw3ZS1g+AsG3bNQG4leV+bFv3uwB+A+Dn4DmWuXza1rkCthaeBIAP61xt6/8cthEtcgCTwG40\ni8Nt5Abbx+ExxToXz7MfzSIC8A6AOtiGs60FIOnw/HWwnVBrhm2ESYwLMgXB9gHT0JbrNwDXd1pm\nNoALbUVrBwANg233V3QYzcIqV9v22te2rfQAsgE82GkZFvsxGrZvVEbYvgLbf+5lmavDvuM7/fyV\nda62dWsAbG87hi4AmO2qdfd0GzHcb10eUyz3G8/zdG8WQgjxBsyvACWEEHLpqJgTQogXoGJOCCFe\ngIo5IYR4ASrmhBDiBaiYE0KIF6BiTgghXoCKOSGEeAEq5oQQ4gX+H63GI20O/smMAAAAAElFTkSu\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f3f5e0e4860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the real minimum and the distribution of the min of a replicate data set\n",
    "plt.figure()\n",
    "plot_tools.modify_axes.only_x(plt.gca())\n",
    "plt.plot(\n",
    "    x,\n",
    "    pdf_min,\n",
    "    label='distribution of the minimum of a replicated data set'\n",
    ")\n",
    "plt.ylim([0, plt.ylim()[1]])  # set y base to zero\n",
    "plt.axvline(\n",
    "    y.min(),\n",
    "    color='k',\n",
    "    linestyle='--',\n",
    "    label='minimum of the true data set'\n",
    ")\n",
    "plt.legend(loc='lower center', bbox_to_anchor=(0.5, 1.05));"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}
